# calculate the absolute value for z is 3z -5i + 1 = z -4i

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Given the equation:

3z - 5i + 1 = z - 4i

We need to find l z l.

First we need to write z into the complex number format z = a+ bi.

==> Then, we will isolate z on the left side.

==> 3z -z = -4i + 5i -1

==> 2z = i -1

Now we will divide by2.

==> z = (-1+i)/2

==> z = ( -1/2) + (1/2)i

Now we will calculate the absolute values.

We know that:

lzl = sqrt(a^2 + b^2)

==> lzl = sqrt(1/2)^2 + (1/2)^2

= sqrt(1/4 + 1/4)

= sqrt( 2/4)

= sqrt(1/2)

**==> lzl = 1/sqrt2 = sqrt2/2**

We first solve for z in the given equation:

3z -5i + 1 = z -4i.

We isolate z to left:

=> 3z-z = -4i+5i-1

=> 2z = -1+i.

=> z = (-1+i)/2.

=> (-1/2)+(1/2)i.

To find absolute z = |z| = square root {(-1/2)^2+(1/2)^2} = (1/2){2)} = (2^(1/2))/2.

|z| = (2^(1/2))/2.